A Novel Explicit Positivity-Preserving Finite-Difference Scheme for Simulating Bounded Growth of Biological Films

In this paper, a novel explicit finite-difference (FD) method is presented to approximate the positive and bounded growth of biological films that is governed by partial differential equations (PDEs) with a nonlinear density-dependent diffusion reaction rate. In such a unique nonlinear equation, the diffusion operator degenerates for small biomass densities, and becomes singular when approaching the upper bound of the biomass density. Our novel FD scheme is designed to transfer the nonlinear terms in the PDE into linear ones that can be solved very efficiently, while ensuring the bounds of the solution between zero and one. This is achieved through (1) a proper design of intertwined FD approximations for the diffusion function, time and spatial variations and (2) the control of the time-step based on stability criteria. A theoretical stability analysis is conducted and it reveals that our method is indeed stable (bounded and not oscillations) in suitable grid spacing and time-steps. The present scheme is applicable for solving convection-diffusion PDEs with variable coefficients, including but not limited to biofilm growth.